On the Bieri–Neumann–Strebel–Renz Σ-invariants of the Bestvina–Brady groups

Abstract

Abstract We study the Bieri–Neumann–Strebel–Renz invariants and we prove the following criterion: for groups H and K of type F ⁢ P n {FP_{n}} such that [ H , H ] ⊆ K ⊆ H {[H,H]\subseteq K\subseteq H} and a character χ : K → ℝ {\chi:K\to\mathbb{R}} with χ ⁢ ( [ H , H ] ) = 0 {\chi([H,H])=0} , we have [ χ ] ∈ Σ n ⁢ ( K , ℤ ) {[\chi]\in\Sigma^{n}(K,\mathbb{Z})} if and only if [ μ ] ∈ Σ n ⁢ ( H , ℤ ) {[\mu]\in\Sigma^{n}(H,\mathbb{Z})} for every character μ : H → ℝ {\mu:H\to\mathbb{R}} that extends χ. The same holds for the homotopical invariants Σ n ⁢ ( - ) {\Sigma^{n}(-)} when K and H are groups of type F n {F_{n}} . We use these criteria to complete the description of the Σ-invariants of the Bieri–Stallings groups G m {G_{m}} , and more generally to describe the Σ-invariants of the Bestvina–Brady groups. We also show that the “only if” direction of the above criterion holds if we assume only that K is a subnormal subgroup of H, where both groups are of type F ⁢ P n {FP_{n}} . We apply this last result to wreath products.